Train
The train is generally the fastest way to get between large cities. The Eurostar is the country's fast train; it takes you from Florence to Rome in 1.5 hours (cost 32 euros), between florence and bologna in 50 minutes, Rome to Naples in 2 hours.
Train schedules, tickets and pricing are available online from Trenitalia.

Validation: Remember that you must always validate the ticket before getting on the train. There are little yellow boxes along the entry to each track. Stamp your ticket here. You will be charged a fine if you forget. This is the way they can tell if you have used the ticket.

Online ticketing of trains: While tickets can be purchased online, there are a lot of limitations of which you must be aware with the electronic ticketing system. It is still rather new and is not at all flexible. If you have an inkling of doubt about taking the train you book, I advise you to book in person at any train station, rather than online. The reason is that online tickets can only be changed online. If you miss your train, you can change for another ticket within one hour, after which it is actually non-refundable. Similarly, if the train is late and you want to take an earlier one or otherwise change your plans, you cannot. You cannot change tickets purchased online in person at the station.

Buying train tickets:
Tickets for Eurostar must be reserved in advance and you are assigned a seat. The car (carrozza) and seat (sedia) number are recorded on the ticket and you must find that place on the train. You can usually buy tickets any day before you travel, in person, at any train station ticket desk, or using the yellow machines in the stations (these accept cash and credit). You can also buy tickets for same-day travel, as long as it is not a major holiday (when you risk not getting a seat). You will need to know which train you want - the machine will give you options for booking, or the person at the window will help you choose. The Eurostar runs mostly on time. If it is more than 30 minutes late, or if you travel without air conditioning in the summer, you have right to a "bonus" (refund). Apply for this at the customer service office at the station of arrival by taking a form which you fill out and post. The bonus will be mailed to you. It takes about a month, so is not a great deal for tourists.
Tickets for Inter-City trains can also be purchased at any time. You do not need to reserve seats, although you can optionally reserve seats. it is best to reserve on major routes like milan-naples or florence-bologna, or on national holidays. These trains make a few more stops than the Eurostar, and hence take longer, but cost proportionally less.
Regional trains are the slowest of the bunch and stop at virtually every little town on the way. While major cities are served by the other types of trains, regional trains serve smaller towns as well as some anomolies like Pisa. Tickets for these trains, which cost really little, can be purchased any time before travel; just remember to validate it before taking the train. Regional trains are almost infallibly late. In the winter they can also be uncomfortably freezing cold.

Services: There are bathrooms on all trains including the regional ones. They are usually disgustingly dirty though sometimes come equipped with toilet paper. Bring your own paper and use hand disinfectant when you're done and you'll survive the experience.

There is a new low cost train service (at present only Rome-Florence-Milan): See the TrenOK website! This works on the low-cost airline business model, with 9 euro bookings in advance up to a certain availability. The trains stop at the less-frequented stations of each city. From these stations, for example firenze campo di marte, you can take another local train between this and the main station, or a city bus to your final destination.